Average Error: 0.0 → 0.0
Time: 12.3s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}
double f(double x, double y) {
        double r9148026 = x;
        double r9148027 = y;
        double r9148028 = 1.0;
        double r9148029 = r9148026 * r9148027;
        double r9148030 = 2.0;
        double r9148031 = r9148029 / r9148030;
        double r9148032 = r9148028 + r9148031;
        double r9148033 = r9148027 / r9148032;
        double r9148034 = r9148026 - r9148033;
        return r9148034;
}
↓
double f(double x, double y) {
        double r9148035 = x;
        double r9148036 = y;
        double r9148037 = 2.0;
        double r9148038 = r9148035 / r9148037;
        double r9148039 = 1.0;
        double r9148040 = fma(r9148038, r9148036, r9148039);
        double r9148041 = r9148036 / r9148040;
        double r9148042 = r9148035 - r9148041;
        return r9148042;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}}\]

  3. Final simplification0.0

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))